Originally Posted by

**Mauritzvdworm** Define for each $\displaystyle \lambda\in\mathbb{R}$ a natural action on $\displaystyle T_{\alpha}$ by rotation:

$\displaystyle

\tau(\lambda)(f)(t) = \left\{

\begin{array}{lr}

f(t+\lambda) & : t+\lambda \in [0,1]\\

\alpha(f(t+\lambda-n)) & : t+\lambda \in [n,n+1],n\in\mathbb{Z}

\end{array}

\right.

$

where $\displaystyle \alpha$ is an automorphism of the $\displaystyle C^*$-algebra A. Prove that the one parameter family $\displaystyle \lambda\mapsto \tau(\lambda)$ gives rise to an isomorphism

$\displaystyle

T_{\alpha}\times_{\alpha}\mathbb{R}\cong (A\times_{\alpha}\mathbb{Z})\otimes \mathbb{K}(L^{2}(S^1))

$

where $\displaystyle T_{\alpha}:=\{f:[0,1]\rightarrow A : f(1)=\alpha(f(0))\}$